Propositiones ad Acuendos Juvenes/Problems/17 - De Tribus Fratribus Singulas Habentibus Sorores
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Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $17$
- De Tribus Fratribus Singulas Habentibus Sorores
- Three Friends and their Sisters
- Three friends, each with a sister, needed to cross a river.
- Each one of them coveted the sister of another.
- At the river, they found only a small boat, in which only two of them could cross at a time.
- How did they cross the river, without any of the women being defiled by the men?
Solution
The constraint here is that at no time may a man be on the same bank as (or in the boat with) the sister of another, without the brother of that sister being present as well.
Let $M_1$, $M_2$ and $M_3$ denote the men, and $S_1$, $S_2$ and $S_3$ denote their respective sisters.
Let them be crossing from $A$ to $B$.
As follows:
Solution 1
- $M_1$ and $S_1$ cross from $A$ to $B$.
- $M_1$ crosses back alone from $B$ to $A$.
- $S_2$ and $S_3$ cross from $A$ to $B$.
- $S_1$ crosses back alone from $B$ to $A$.
- $M_2$ and $M_3$ cross from $A$ to $B$.
- $M_2$ and $S_2$ cross back from $B$ to $A$.
- $M_1$ and $M_2$ cross from $A$ to $B$.
- $S_3$ crosses back alone from $B$ to $A$.
- $S_1$ and $S_3$ cross from $A$ to $B$.
- $M_3$ crosses back alone from $B$ to $A$.
- $M_3$ and $S_3$ cross from $A$ to $B$.
- "And that would complete the crossing without anything untoward happening."
This was the original solution given in Propositiones ad Acuendos Juvenes by Alcuin of York.
It is not optimal.
$\blacksquare$
Solution 2
- $M_1$ and $S_1$ cross from $A$ to $B$.
- $M_1$ crosses back alone from $B$ to $A$.
- $S_2$ and $S_3$ cross from $A$ to $B$.
- $S_1$ crosses back alone from $B$ to $A$.
- $M_1$ and $S_1$ cross from $A$ to $B$.
- $M_1$ crosses back alone from $B$ to $A$.
- $M_2$ and $M_3$ cross from $A$ to $B$.
- $S_1$ crosses back alone from $B$ to $A$.
- $M_1$ and $S_1$ cross from $A$ to $B$.
$\blacksquare$
Also presented as
This can also be presented as men with their wives, rather than sisters.
Sources
- c. 800: Alcuin of York: Propositiones ad Acuendos Juvenes ... (previous) ... (next)
- 1992: John Hadley/2 and David Singmaster: Problems to Sharpen the Young (Math. Gazette Vol. 76, no. 475: pp. 102 – 126) www.jstor.org/stable/3620384
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): 'Propositions to Sharpen Up the Young': $81$